(This entry describes two distinct notions, one in the theory of inner product spaces, and the second in a more purely category theoretic context.)
Two elements in an inner product space, , are orthogonal or normal vectors, denoted if .
Two morphisms and in a category are said to be orthogonal, written , if has the left lifting property with respect to , i.e. if in any commutative square
there exists a unique diagonal filler making both triangles commute:
Given a class of maps , the class is denoted or . Likewise, given , the class is denoted or . These operations form a Galois connection on the poset of classes of morphisms in the ambient category. In particular, we have and .
A pair such that and is sometimes called a prefactorization system. If in addition every morphism factors as an -morphism followed by an -morphism, it is an (orthogonal) factorization system.
Of course, any orthogonal factorization system gives plenty of examples. The ur-example is that in Set (or actually, any pretopos) for any surjection and injection .
A strong epimorphism in any category is, by definition, an epimorphism in , where is the class of monomorphisms. (If the category has equalizers, then every map in is epic.) Dually, a strong monomorphism is a monomorphism in .
The orthogonal subcategory problem for a class of morphisms in a category asks whether the full subcategory of objects orthogonal to is a reflective subcategory. Here we define to mean .
The orthogonal subcategory problem is related to localization. Suppose is indeed a reflective subcategory; let be the reflector (the left adjoint to the inclusion ). Certainly sends arrows in to isomorphisms in . Indeed, if belongs to , then the inverse to is the unique arrow extending along to an arrow , using the fact that belongs to .
type of subspace of inner product space | condition on orthogonal space | |
---|---|---|
isotropic subspace | ||
coisotropic subspace | ||
Lagrangian subspace | (for symplectic form) | |
symplectic space | (for symplectic form) |
Last revised on October 4, 2023 at 15:03:21. See the history of this page for a list of all contributions to it.